Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#10831 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #381 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C2×C53 | #1434 |
| prime radical | C24×C53 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C23×C4×C53 ◁ G | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,11,13,7)(4,15)(5,10)(6,9)(8,16)(12,19,25,23)(14,24,27,20)(17,28)(18,22)(21,29)(26,30,32,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,5,14,6,13,18,27)(2,7,10,20,9,11,22,24)(4,12,17,26,16,25,29,32)(8,19,21,30,15,23,28,31);(1,4)(2,8,9,15)(3,12)(5,17)(6,16)(7,19,11,23)(10,21,22,28)(13,25)(14,26)(18,29)(20,30,24,31)(27,32)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3819);